5.1 Near-extremal Kerr–Newman black holes

Near-extremal Kerr–Newman black holes are characterized by their mass M, angular momentum J = M a and electric charge Q. (We take a, Q ≥ 0 without loss of generality.) They contain near-extremal Kerr and Reissner–Nordström black holes as particular instances. The metric and thermodynamic quantities can be found in many references and will not be reproduced here.

The near-extremality condition (145View Equation) is equivalent to the condition that the reduced Hawking temperature is small,

r+-−--r− τH ≡ r ≪ 1. (147 ) +
Indeed, one has τH = M TH [4π((r+∕M )2 + (a∕M )2)∕(r+ ∕M )] and the term in between the brackets is of order one since 0 ≤ a∕M ≤ 1, 0 ≤ Q∕M ≤ 1 and 1 ≤ r+ ∕M ≤ 2. Therefore, we can use interchangeably the conditions (145View Equation) and (147View Equation).

Since there is both angular momentum and electric charge, extremality can be reached both in the regime of vanishing angular momentum J and vanishing electric charge Q. When angular momentum is present, we expect that the dynamics could be described by the CFTJ, while when electric charge is present the dynamics could be described by the CFTQ. It is interesting to remark that the condition

TH-≪ 1, (148 ) ΩJ
implies (145View Equation) – (147View Equation) since τH = TH-(4πa∕M ) ≪ 1 ΩJ but it also implies a > 0. Similarly, the condition
TH- M Φe ≪ 1 (149 )
implies (145View Equation) – (147View Equation), since τH = 4πQTH ∕Φe, but it also implies Q > 0. In the following, we will need only the near-extremality condition (145View Equation), and not the more stringent conditions (148View Equation) or (149View Equation). This is the first clue that the near-extremal scattering will be describable by both the CFTJ and the CFTQ.

Near-extremal black holes are characterized by an approximative near-horizon geometry, which controls the behavior of probe fields in the window (146View Equation). Upon taking TH = O (λ) and taking the limit λ → 0 the near-horizon geometry decouples, as we saw in Section 2.7.

Probes will penetrate the near-horizon region close to the superradiant bound (146View Equation). When TH = O (λ) we need

ext ext ω = m Ω J + qeΦ e + O (λ). (150 )
Indeed, repeating the reasoning of Section 2.6, we find that the Boltzman factor defined in the near-horizon vacuum (defined using the horizon generator) takes the following form
ℏω−m-ΩTJ−-qeΦe −ℏn − ℏmTϕ-−ℏqTee e H = e e , (151 )
where ω, m and qe are the quantum numbers defined in the exterior asymptotic region and
ω-−--m-ΩeJxt−-qeΦexet n ≡ T , (152 ) H
is finite upon choosing (150View Equation). The conclusion of this section is that the geometries (79View Equation) control the behavior of probes in the near-extremal regime (145View Equation) – (146View Equation). We identified the quantity n as a natural coefficient defined near extremality. It will have a role to play in later Sections 5.3 and 5.4. We will now turn our attention to how to solve the equations of motion of probes close to extremality.
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